Commuting graphs of some subsets in simple rings

AbstractLet D be a division ring with center F and n⩾1 a natural number. For S⊆Mn(D) the commuting graph of S, denoted by Γ(S), is the graph with vertex set S⧹Z(S) such that distinct vertices a and b are adjacent if and only if ab=ba. In this paper we prove that if n>2 and A,N,I,T are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, Γ(A), Γ(N), Γ(I), and Γ(T) are connected graphs. We show that if n>2 and U is the set of all upper triangular matrices, then for every algebraic division ring D, Γ(U) is a connected graph. Also it is shown that if R is the set of all reducible matrices and Mn(D) is algebraic over F, then for n>2, Γ(R) is a connected graph. Finally, we prove that for n⩾2, Γ(Mn(H)) is a connected graph, where H is the ring of real quaternions